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Register a new userA Variant of the Wang-Foster-Kakade Lower Bound for the Discounted Setting
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Nan Jiang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Recently, Wang et al. (2020) showed a highly intriguing hardness result for batch reinforcement learning (RL) with linearly realizable value function and good feature coverage in the finite-horizon case. In this note we show that once adapted to the discounted setting, the construction can be simplified to a 2-state MDP with 1-dimensional features, such that learning is impossible even with an infinite amount of data.
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A fundamental question in the theory of reinforcement learning is: suppose the optimal $Q$-function lies in the linear span of a given $d$ dimensional feature mapping, is sample-efficient reinforcement learning (RL) possible? The recent and remarkable result of Weisz et al. (2020) resolved this question in the negative, providing an exponential (in $d$) sample size lower bound, which holds even if the agent has access to a generative model of the environment. One may hope that this information theoretic barrier for RL can be circumvented by further supposing an even more favorable assumption: there exists a emph{constant suboptimality gap} between the optimal $Q$-value of the best action and that of the second-best action (for all states). The hope is that having a large suboptimality gap would permit easier identification of optimal actions themselves, thus making the problem tractable; indeed, provided the agent has access to a generative model, sample-efficient RL is in fact possible with the addition of this more favorable assumption. This work focuses on this question in the standard online reinforcement learning setting, where our main result resolves this question in the negative: our hardness result shows that an exponential sample complexity lower bound still holds even if a constant suboptimality gap is assumed in addition to having a linearly realizable optimal $Q$-function. Perhaps surprisingly, this implies an exponential separation between the online RL setting and the generative model setting. Complementing our negative hardness result, we give two positive results showing that provably sample-efficient RL is possible either under an additional low-variance assumption or under a novel hypercontractivity assumption (both implicitly place stronger conditions on the underlying dynamics model).
We study the reinforcement learning problem for discounted Markov Decision Processes (MDPs) under the tabular setting. We propose a model-based algorithm named UCBVI-$gamma$, which is based on the emph{optimism in the face of uncertainty principle} and the Bernstein-type bonus. We show that UCBVI-$gamma$ achieves an $tilde{O}big({sqrt{SAT}}/{(1-gamma)^{1.5}}big)$ regret, where $S$ is the number of states, $A$ is the number of actions, $gamma$ is the discount factor and $T$ is the number of steps. In addition, we construct a class of hard MDPs and show that for any algorithm, the expected regret is at least $tilde{Omega}big({sqrt{SAT}}/{(1-gamma)^{1.5}}big)$. Our upper bound matches the minimax lower bound up to logarithmic factors, which suggests that UCBVI-$gamma$ is nearly minimax optimal for discounted MDPs.
A random net is a shallow neural network where the hidden layer is frozen with random assignment and the output layer is trained by convex optimization. Using random weights for a hidden layer is an effective method to avoid the inevitable non-convexity in standard gradient descent learning. It has recently been adopted in the study of deep learning theory. Here, we investigate the expressive power of random nets. We show that, despite the well-known fact that a shallow neural network is a universal approximator, a random net cannot achieve zero approximation error even for smooth functions. In particular, we prove that for a class of smooth functions, if the proposal distribution is compactly supported, then a lower bound is positive. Based on the ridgelet analysis and harmonic analysis for neural networks, the proof uses the Plancherel theorem and an estimate for the truncated tail of the parameter distribution. We corroborate our theoretical results with various simulation studies, and generally two main take-home messages are offered: (i) Not any distribution for selecting random weights is feasible to build a universal approximator; (ii) A suitable assignment of random weights exists but to some degree is associated with the complexity of the target function.
Modern tasks in reinforcement learning have large state and action spaces. To deal with them efficiently, one often uses predefined feature mapping to represent states and actions in a low-dimensional space. In this paper, we study reinforcement learning for discounted Markov Decision Processes (MDPs), where the transition kernel can be parameterized as a linear function of certain feature mapping. We propose a novel algorithm that makes use of the feature mapping and obtains a $tilde O(dsqrt{T}/(1-gamma)^2)$ regret, where $d$ is the dimension of the feature space, $T$ is the time horizon and $gamma$ is the discount factor of the MDP. To the best of our knowledge, this is the first polynomial regret bound without accessing the generative model or making strong assumptions such as ergodicity of the MDP. By constructing a special class of MDPs, we also show that for any algorithms, the regret is lower bounded by $Omega(dsqrt{T}/(1-gamma)^{1.5})$. Our upper and lower bound results together suggest that the proposed reinforcement learning algorithm is near-optimal up to a $(1-gamma)^{-0.5}$ factor.
We derive a novel asymptotic problem-dependent lower-bound for regret minimization in finite-horizon tabular Markov Decision Processes (MDPs). While, similar to prior work (e.g., for ergodic MDPs), the lower-bound is the solution to an optimization problem, our derivation reveals the need for an additional constraint on the visitation distribution over state-action pairs that explicitly accounts for the dynamics of the MDP. We provide a characterization of our lower-bound through a series of examples illustrating how different MDPs may have significantly different complexity. 1) We first consider a difficult MDP instance, where the novel constraint based on the dynamics leads to a larger lower-bound (i.e., a larger regret) compared to the classical analysis. 2) We then show that our lower-bound recovers results previously derived for specific MDP instances. 3) Finally, we show that, in certain simple MDPs, the lower bound is considerably smaller than in the general case and it does not scale with the minimum action gap at all. We show that this last result is attainable (up to $poly(H)$ terms, where $H$ is the horizon) by providing a regret upper-bound based on policy gaps for an optimistic algorithm.
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